To compare Theorem REF  and REF , the lower bound for sparse model is much smaller than the one for dense model. This indicates that there is a potential improvement for adversarially robust estimators if the algorithm can utilize the sparsity information (if there is). As discussed in [1]}, [2]}, for high-dimensional model, if we do not consider the sparsity information, the resulting model is not consistent in both standard and adversarially robust learning problems.
